Monday, 20 Sep 2021

Illustrating the Need for an Axiomatic Structure of a Mathematical System in General

A line is a –dimensional figure that contains a set of points arranged in a row that extended infinitely in both directions. Two points determine a line. That is, two distinct points are contained by exactly one line. A line can be named using a lower case letter or any two points on the line.

Direction:  Write TRUE if the statement is correct and FALSE if the statement is wrong. Use the figure below for the given item.

                                                                                      E

  1. Points A, B, C, D are collinear.
  2. Points A, D, F are noncollinear
  3. Points B, F and G are on the same line 4. Points G, C, D are not on the same line.
  4. Points A, E, F are coplanar.
  5. Points A, F, G are not coplanar. .
  6. Points A, B, D , E are on the same plane.
  7. Points A, B, F, E are coplanar.
  8. Points A. B, D are collinear and coplanar.
  9. Points B, F, C are collinear and coplanar.

Ans. Exercise 2:

1. FALSE

2. TRUE

3. FALSE

4. TRUE

5. TRUE

6. TRUE

7. FALSE

8. TRUE

9. FALSE

10. FALSE

Exercise 3

Direction: Illustrate each of the following and label the diagram.

  1. Point A lies in plane P.
  2. Plane M contains line AB.
  3. If two angles form a linear pair, then they are supplementary.
  4. Vertical angles are congruent.

 (ANS ON SEPARATE SHEET)

Exercise 4

Direction: Complete the table.

.

 NameDescription or meaning  Illustration or Figure
Undefined term Three undefined terms of geometry the term undefined is often used to refer to an expression which is not assigned an interpretation or a value (such as an indeterminate form, which has the propensity of assuming different values). The term can take on several different meanings depending on the context. The three undefined terms are point, line, and plane. Thus, figure D represents an undefined term as it’s a line.
  Defined term   A formal definition. A formal definition and can be defined using other geometrical terms. We defined the other terms of the mathematical system in terms of undefined terms. angle, line segment , circle
  Postulate   statement that is accepted as true A postulate is a statement that is accepted as true without having to formally prove it. a well-known postulate in mathematics is the segment addition postulate, which states the following: Segment Addition Postulate:If a point, B, is drawn on a line segment AC, then AC is the sum of AB and BC.
  Theorem   A general proposition not self-evident By a chain of reasoning; a truth established by means of accepted truths. In general, a theorem is an embodiment of some general principle that makes it part of a larger theory. The process of showing a theorem to be correct is called a proof. Linear Pair Theorem If two angles form a linear pair, then they are supplementary

Exercise 5

Direction: Match the statements from column A to its corresponding terms in                       column B.

                             Column A                                     Column B
1. If two angles form a linear pair,            a. Point           
     then they are supplementary.                        
2. Vertical angles are congruent.             b. Angle Addition Postulate           
3. If point S lies in the interior of ∠PQR, c. Line                      
     then m∠PQS + m∠SQR = m∠PQR.                                 
4. If points P, Q, and R are collinear            (P–Q–R) and Q is between points d. Segment Addition Postulate
    P and R, then ̅PQ̅̅̅̅ + ̅QR̅̅̅̅ = ̅PR̅̅̅.             e. Linear Pair Theorem
5. If ̅QS̅̅̅̅ bisects ∠PQR,         then ∠PQS ≅ ∠SQR   f. Definition of an Angle Bisector
It suggests an exact location        in space.  They are points found on theg. Vertical Angles Theorem
     same line.                                         h. Coplanar points
It is a set of points in an                     endless flat surface.                              They are points found on i. plane                     
     the same plane.                                  10. It is a one –dimensional figure 
      that contains set of points arranged         in a row which extended       infinitely in both directions.   Ans in Exercise 5: 1. E 2. G 3.B 4.D 5.F 6.A 7.J 8.i 9, H 10. C                  j. Collinear points

Questions:

  1. Did you encounter any difficulties in illustrating the undefined and defined terms, postulates and theorems in Geometry? If yes, please specify. 

Yes,  such as –dimensional figure that contains set of points arranged

in a row which extended infinitely in both directions.

  • How did you illustrate the undefined and defined terms, postulates and theorems in Geometry?

I illustrate the undefined and defined terms, postulates and theorems in Geometry using the given label and the name of the lines, angles, and plane.

  • Formulate your own real-life situation which illustrates postulates or theorems.

Now, are you going to accept her statement as true, or are you going to whip out a tape measure and measure the length of everyone’s hair to verify the truth of her statement? Most likely, you would accept her statement as true, because it’s fairly obvious that she has the longest hair in the group.

Image: Pexels

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